1. Field of the Invention
The present invention generally relates to method, system and software product used in structural design and analysis using finite element analysis, more particularly to control zero-energy hourglass mode deformations of solid elements in finite element analysis.
2. Description of the Related Art
Finite element analysis (FEA) is a computerized method widely used in industry to model and solve engineering problems relating to complex systems. FEA derives its name from the manner in which the geometry of the object under consideration is specified. With the advent of the modern digital computer, FEA has been implemented as FEA software. FEA software can be classified into two general types, implicit FEA software and explicit FEA software. Implicit FEA software uses an implicit equation solver to solve a system of coupled linear equations. Such software is generally used to simulate static or quasi-static problems. Explicit FEA software does not solve coupled equations but explicitly solves for each unknown assuming them uncoupled. Explicit FEA software usually uses central difference time integration which requires very small solution cycles or time steps for the method to be stable and accurate. Explicit FEA software is generally used to simulate short duration events where dynamics are important such as impact type events.
The user of FEA software creates a model of the system to be analyzed using elements. An element represents a finite region of the system. Within each element, the unknown quantity is assumed to take a simple form within the domain of the element. For explicit FEA software, the unknown quantity is usually acceleration. For implicit FEA software, the unknown quantity may be displacement, velocity, temperature, or others.
By assuming a simple form of the unknown within an element, and by using many elements, complex behaviors can be simulated within a reasonable time frame with the FEA software. One simple example is a linear spring element with the displacement at the ends of the spring as the unknown quantity. The displacement field is assumed to vary linearly along the length of the spring. Therefore, if we solve for the displacement at the ends of the spring, we can easily evaluate the displacement at any point along the spring. The compatible strain and stress fields for a linear spring element are constant over the element length and are easily evaluated from the end displacements and material properties.
The points on the element where the unknown is solved are called nodes. The linear spring has a node at each end. If we place a third node at the middle of the spring element, the displacement field could then be assumed to vary as a quadratic function and the compatible strain and stress fields would be linear. The common element types are solid elements for modeling volumes, shell elements for modeling thin parts dominated by bending, beam elements for modeling beams, and spring or truss elements for modeling springs and trusses. Each element is assigned a material type and appropriate material properties. By choosing appropriate material types and properties, metals, plastics, foams, soil, concrete, rubber, glass, fluids and many other materials can be modeled. The user must also specify the boundary conditions, loads, and initial conditions to complete the model. To accurately simulate complex system behavior, many elements are needed. Today, typical FEA models of entire automobile are made of more than 500,000 elements.
Once the model is defined, FEA software can perform a simulation of the physical behavior under the specified loading or initial conditions. FEA software is used extensively in the automotive industry to simulate front and side impacts of automobiles, occupant dummies interacting with airbags, and the forming of body parts from sheet metal. Such simulations provide valuable insight to engineers who are able to improve the safety of automobiles and to bring new models to the market more quickly.
Solid elements are typically used for modeling thick parts or solid bodies. In three dimensions, a solid element can be shaped like brick or hexahedron. The lowest order brick element has a node at each corner and is thus called the 8-node brick or hexahedral element. The 8-node brick element can be assumed to have a displacement (or other unknown) field that varies linearly along the edges between the nodes. Throughout the element domain, the displacement field has linear terms and cross terms but no quadratics or higher order terms. The compatible stress and strain fields have linear terms within the element domain. There are other types of solid elements such as the 6-node pentahedral element.
To calculate the nodal forces that are generated by the stress within an element, the calculated stress tensor is multiplied by terms that account for the element's geometry and then integrated over the domain of the element. For many materials, the stress field does not have a linear relationship to the strain field, so closed form integration is not possible. Instead, numerical integration such as Gauss-Legendre quadurature is routinely used. Numerical integration of an 8-node brick element can be done by defining two Gauss-Legendre integration points in each spatial direction for a total of 8 integration points. Such element is said to have full integration or rank sufficient integration. Full integration guarantees that all possible modes of deformation generate stress in the element. Alternatively, numerical integration of an 8-node brick element can be done with a single Gauss-Legendre integration point. Such an element is called an under-integrated or rank deficient element.
The under-integrated element is faster than the fully integrated element because only one stress evaluation is required per element per solution cycle rather than eight in a fully integrated element. This speed advantage is very noticeable in an explicit FEA solution where the stress evaluation dominates the solution time. An under-integrated element is also more robust in large deformation calculations because the terms evaluated at the integration point remain well conditioned at larger deformation. This follows directly from the location of the single point, at the element center, rather than shifted towards the corners as with the fully integrated element.
The disadvantage to numerical integration with only one integration point is that some modes of deformation do not contribute to the strain or stress field at the integration point. These modes which were originally noticed in finite difference calculations in two dimensions in the 1960's are historically called hourglass modes because of their shape as shown in FIGS. 1A and 1B. These modes are also called zero-energy modes because they do not generate any strain energy in the element. FIG. 2A shows an 8-node brick element with one integration point in the center of the element. The 8-node element has four zero-energy modes in each of three spatial directions for a total of 12. The four modes in one of the spatial directions are shown in FIG. 2B.
To use under-integrated elements in a finite element analysis, it is necessary to limit the deformation due to the zero-energy modes. Otherwise, the deformation due to these zero-energy modes may grow too large and dominate the solution. Some existing methods for controlling hourglass deformation are found in the following papers: “A Uniform Strain Hexahedron and Quadrilateral with Orthogonal Hourglass Control” by Flanagan and Belytsckko (1981); “Efficient Implementation of Quadrilaterals with High Coarse-Mesh Accuracy” by Belytschko and Bachrach (1985); and “Assumed Strain Stabilization of the Eight Node Hexahedral Element” by Belytschko and Bindeman (1993).
To be accurate for large deformation analysis, these existing hourglass control methods calculate an increment of hourglass force using the current element geometry to scale the force increments. The force increments are summed over time to obtain the total hourglass force. Because the element geometry changes during the solution, the increments of hourglass force calculations are based on a variable stiffness. This nonlinearity in the stiffness can cause residual hourglass deformation to remain in an element after some load is applied to the element and then removed. One exemplary application that demonstrates residual hourglass deformation is an automobile tire model that was used to study its behavior during skidding. The tire model was first pressed onto the road with a force to represent the weight of the vehicle. The tire model was then moved with a prescribed motion to start the tire rolling. After some time, the prescribed motion caused the rolling tire to skid. 8-node brick elements were used to model the tire tread. If we observe one element that models a small part of the tire tread, we would see it rotating with the tire. At some point, the element would contact the road and be compressed by the weight of the automobile. Because the tire is skidding, we would also see frictional forces cause shear deformation of the element. After a few moments, the rotation of the tire would cause the deformed element to separate from the road surface. With the compressive and shear loading removed, we would expect the element to spring back to its undeformed geometry. However, after a few rotations of the tire, we observed the brick elements of the tread failing to spring back to their undeformed geometries.
It is therefore desirable to have a new method to control hourglass deformations for solid elements in the applications that the traditional methods do not work well.